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NOVEL TECHNIQUES FOR PULSED FIELD GRADIENT NMR MEASUREMENTS By WILLIAM W. BREY A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1994 ACKNOWLEDGMENTS I would like to thank Janel LeBelle, Igor Friedman, and Don Sanford for construction of the gradient coil prototypes, and Jerry Dougherty for performing the simulations of the coils they all helped to construct. Stanislav Sagnovski, Eugene Sczezniak, Doug Wilken, and Randy Duensing participated in many helpful discussions concerning gradient coils. Debra NeillMareci provided the excellent illustration of a gradient coil in Figure 22. For their part in the microscopy project, thanks go to Barbara Beck, Michael Cockman, and Dawei Zhou. Ed Wirth and Louis Guillette provided the samples. For help measuring eddy current fields I am grateful to Wenhua Xu, and to Steve Patt for help with the software. Thanks go to my parents, Mary Louise and Wallace Brey, and my brother, Paul Brey, for encouragement and help with red tape. Paige Brey has my special thanks for her extensive help preparing the thesis. Katherine Scott, Richard Briggs, Jeff Fitzsimmons, and Neil Sullivan enriched my graduate experience with their wide knowledge and diverse interests. Thanks go to them for their enthusiasm and for reading this thesis. Raymond Andrew served as supervisory committee chairman. Thanks go to Thomas Mareci for directing the research, for providing financial and moral support, and for encouraging me to pursue this work to its conclusion. iii TABLE OF CONTENTS ACKNOWLEDGMENTS .......................................... ii ABSTRACT ................................................. v GENERAL INTRODUCTION ....................................... 1 MEASUREMENT OF EDDY CURRENT FIELDS ....................... 5 Introduction ........................................ 5 Literature Review ................................... 11 SpinEcho Techniques ................................ 23 Stimulated Echo Techniques .......................... 28 Results ............................................. 34 Conclusion .......................................... 41 GRADIENT COIL DESIGN ..................................... 46 Introduction and Theory ............................. 46 Literature Review ................................... 50 Field Linearity ..................................... 61 Efficiency .......................................... 62 Eddy Currents ....................................... 68 Coil Projects ....................................... 72 Amplifiers ....................................... 72 16 mm Coil for NMR Microscopy .................. 76 9 cm Coil for Small Animals .................... 79 15 cm Coil for Small Animals ................... 87 Concentric Return Path Coil .................... 98 SYSTEM DEVELOPMENT FOR NMR MICROSCOPY .................... 126 Introduction ........................................ 126 Literature Review ................................... 128 Instrument Development .............................. 133 Results ............................................. 149 Conclusion .......................................... 153 CONCLUSION ............................................... 155 REFERENCES ............................................... 156 BIOGRAPHICAL SKETCH ...................................... 162 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NOVEL TECHNIQUES FOR PULSED FIELD GRADIENT NMR MEASUREMENTS By William W. Brey December, 1994 Chairman: E. Raymond Andrew Major Department: Physics Pulsed field gradient (PFG) techniques now find application in multiple quantum filtering and diffusion experiments as well as in magnetic resonance imaging and spatially selective spectroscopy. Conventionally, the gradient fields are produced by azimuthal and longitudinal currents on the surfaces of one or two cylinders. Using a series of planar units consisting of azimuthal and radial current elements spaced along the longitudinal axis, we have designed gradient coils having linear regions that extend axially nearly to the ends of the coil and to more than 80% of the inner radius. These designs locate the current return paths on a concentric cylinder, so the coils are called Concentric Return Path (CRP) coils. Coils having extended linear regions can be made smaller for a given sample size. Among the advantages that can accrue from using smaller coils are improved gradient strength and switching time, reduced eddy currents in the absence of shielding, and improved use of bore space. We used an approximation technique to predict the remaining eddy currents and a timedomain model of coil performance to simulate the electrical performance of the CRP coil and several reduced volume coils of more conventional design. One of the conventional coils was designed based on the timedomain performance model. A singlepoint acquisition technique was developed to measure the remaining eddy currents of the reduced volume coils. Adaptive sampling increases the dynamic range of the measurement. Measuring only the center of the stimulated echo removes chemical shift and B0 inhomogeneity effects. The technique was also used to design an inverse filter to remove the eddy current effects in a larger coil set. We added pulsed field gradient and imaging capability to a 7 T commercial spectrometer to perform neuroscience and embryology research and used it in preliminary studies of binary liquid mixtures separating near a critical point. These techniques and coil designs will find application in research areas ranging from functional imaging to NMR microscopy. GENERAL INTRODUCTION As pulsed field gradient technology for NMR matures, new and diverse applications develop. Pulsed Gradient Spin Echo techniques allow the measurement not only of the bulk diffusion tensor, but of the structure factor of the sample.1 Editing techniques use pulsed field gradients to simplify the complex spectra of biomolecules.2 Local gradient coils allow functional imaging in the human head.3 NMR microscopy can require field gradients much larger and switched more rapidly than conventional imaging experiments.4 Localized spectroscopy allows chemical shift information to be collected from specific voxels in a living animal.5 This paper will address some approaches for producing and evaluating pulsed field gradients. A technique was developed to measure the eddy current field that persists after a field gradient is switched off and, based on the measurement, a filter to correct for the eddy current field was designed. The technique, which employs a series of experiments based on the stimulated echo, was then used to evaluate the performance of the 1D. G. Cory and A. N. Garroway, Magn. Reson. Med. 14, 435, 1990. 2D. Brihwiler and G. Wagner, J. Magn. Reson. 69, 546, 1986. 3K. K. Kwong et al., Proc. Natl. Acad. Sci. 89, 5675, 1992. 4Z. H. Cho et al., Med. Phys. 15, 815, 1988. 5H. R. Brooker et al., Macn. Reson. Med. 5, 417, 1987. filter. If an eddy current field persists during the period when the NMR signal is detected, distortions in the spectrum or image will result. The distortions are particularly severe when chemical shift information is obtained in the same experiment as spatial localization by encoding spatial information in the phase of the NMR signal. It is important to be able to measure the residual gradient field, which is usually due to eddy currents in the metal structures of the magnet, so that it can be corrected by changing the drive to the gradient amplifier, or by whatever other technique is available, and to evaluate the remaining uncorrected field to estimate the distortion that will result in a desired experiment. One way to avoid eddy currents for experiments such as spatially selective spectroscopy is to employ actively shielded gradient coils. Another, much simpler, approach is to reduce the size of the gradient coil so that it is widely separated from the eddycurrentproducing structures in the magnet. This approach is only possible when the clear bore of the magnet is much larger than the volume of interest, which is often the case. To make possible experiments, such as spatially selective spectroscopy, that require rapidly switched high intensity field gradients, I developed pulsed field gradient systems based on reduced volume gradient coils for a 2 T, 31 cm bore magnet used for small animal studies. This magnet was replaced with a 4.7 T, 33 cm bore magnet, and the gradient systems were adapted accordingly. These pulsed field gradient systems offer much better performance than the large and unshielded gradient system supplied with the magnet, given their limitation on sample size. I also developed a pulsed field gradient coil for a 7 T, 51 mm bore magnet used for NMR microscopy and spectroscopy. Another experiment which requires gradient coils to perform exceptionally well is functional imaging of the human brain. The head is much smaller than the wholebody magnets in general use. A smaller coil can allow faster switching to higher gradient fields, as well as reduce eddy current fields. In order to get a gradient coil that is matched to the size of the head, some provision must be made to allow for the shoulders. Conventional designs, even existing designs with a large linear volume, have current return paths arrayed on both sides of the linear volume. A coil matched to the size of the head would not fit over the shoulders. A coil that trades radial linear region for increased axial linear region is more appropriate. A design utilizing concentric return paths was developed that significantly improved the axial region of linearity. A prototype was constructed and tested. In order to perform NMR microscopy and pulsed field gradient experiments, we adapted an NMR spectrometer and probe for a 7 T, 51 mm bore magnet. The instrument included a simple amplitude modulator to carry out slice selection. A probe that allowed sample loading from above was 4 constructed. Artifacts were eliminated from the images. A software interface that allows the user to set up an experiment by entering values in a spreadsheet was developed. Useful contrast was obtained on fixed biological samples. Preliminary imaging experiments on both biological and nonbiological systems were carried out. MEASUREMENT OF EDDY CURRENT FIELDS Introduction It is well known that, when a current pulse is passed through a field gradient coil in a superconducting magnet, eddy currents are produced in the conducting structures of the magnet. Experiments such as diffusionweighted imaging6 and multiplequantum spectroscopy7 require that the eddy current field be a much smaller fraction of the applied field than do conventional spinecho magneticresonance imaging experiments. Strategies to reduce the eddy current field consequently become increasingly important. The two effective strategies are signal processing of the gradient demand, known as preemphasis, and selfshielding of gradient coils, which greatly reduces the interaction of the coil with the metal structures of the magnet. Often, the two techniques are used together. When the sample or subject is substantially smaller than the magnet, another approach is to minimize the size of the gradient coil. In order to evaluate and improve the effectiveness of these three strategies, it is desirable to have a technique to measure eddy current fields. To implement the preemphasis, it is necessary to measure the eddy current field in order to 6D. G. Cory and A. N. Garroway, Magn. Reson. Med. 14, 435, 1990. 7C. Boesch et al., Magn. Reson. Med. 20, 268, 1991. cancel it. An eddy current measurement technique is also useful in order to evaluate the possibility of performing a given experiment with available hardware. In this chapter, a technique for measuring and analyzing the time behavior of eddy current fields is developed and experimental results are presented. Some general physical considerations of eddy currents are discussed, and existing techniques for eddy current field measurement are reviewed. An introduction to the Bloch equations will be preliminary to a discussion of the effect of the eddy current field on the nuclear magnetization. The Bloch equations provide a phenomenological description of some aspects of the behavior of spins in a magnetic field. Let M be the bulk nuclear magnetization, y the gyromagnetic ratio, By the polarizing magnetic field, and B1 the amplitude of the radio frequency excitation field which has rotational frequency o. T1 and T2 are the time constants associated with longitudinal and transverse relaxation, respectively. Mx = Y(BoMy + BlMz sin (ot) Mx/T2 [1] My = Y(BiMZ cos Ct BoMx) My/T2 [2] Mz = y(BlMx sin Cot + BiMy cos Cot) (Mz Mo)/IT [3] Instead of T2, the symbol T2* is used to denote the time constant of apparent transverse relaxation when inhomogeneity in BO is present. Neglecting the effects of T1 and T2 and assuming B1 = 0, the equations can be simplified. MX = TMyBo [4] My = 'YMBO [5] Mz = 0 [6] We can introduce a complex transverse magnetization M = Mx + iMy so that M = iyMB0. [7] Assume that Bo consists of a constant and a component linearly dependent on position: B0 = BO+gx. BO is independent of time and space, while g is quasistatic. If we define m, the magnetization in the rotating frame, by M = meiBOt [8] then M = iyBOM + meiyBot [9] Substituting back into Equation [7] gives iyBOM + ineiyBot = iyM(BO + gx). [10] Simplifying Equation [10] yields ne iyBO t = iyMgx. [11] Combining Equation [11] with Equation [8] yields m = iygxm, [12] which has the immediate solution iyxft gdt' m(t) = m(to)e t [13] If the magnetization has been prepared to a nonzero m(to) by a radio frequency pulse, the evolution described by Equation [13] is called a free induction decay (FID). Consider the characteristics of a general eddy current field. The eddy currents give rise to a magnetic field that roughly tends to cancel the applied field of the gradient coil. The spatial dependence of the eddy current field is not exactly the same as the applied gradient field.8 The time behavior of the eddy current field is a multiexponential decay, which can be seen by considering the form of the solution to the differential equation governing the decay of magnetic induction due to current flow in the conductor. Maxwell's equations9 in a vacuum in SI units are V B = 0 [14] V *E = [15] aB V X E +  = 0 [16] at V X B oF00 DE = o0J [17] where B is the magnetic induction and E is the electric field, p is the charge density, e0 and go are the permittivity and permeability of free space, and J is the current density. We also assume Ohm's law, J = GE, where y is the conductivity, assumed to be isotropic and homogeneous. Taking the curl of both sides of Ampere's law, Equation [17], neglecting the displacement current, and using the identity 8R. Turner and R.M. Bowley, J. Phys E: Sci. Instrum. 19, 876, 1986. 9J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1975. V x (V x A) = V(V .A) V2A [18] gives V2B = g0V X J. [19] Using Ohm's law to eliminate J for E, neglecting the displacement current, yields V2B = 0CFV X E, [20] so Equation [16] allows this to be expressed as rB V2B = 0a [21] at The decay of the magnetic induction must be a solution to this diffusion equation. Separation of variables gives solutions for the time part having an exponential time dependence. This makes it possible to correct for the linear spatial term in the eddy current field with a linear filter network. Such a network is known as a preemphasis circuit. In a superconducting magnet, the conducting structures involved are often at very low temperatures and hence have much greater conductivity than might otherwise be expected. For example, pure aluminum at 10 K has a resistivity of 1.93 x 1012 J2m, while at a room temperature of 293 K its resistivity10 is 2.65 x 108 Qm. The time scale of the eddy current decay is directly proportional to its conductivity, as can be inferred from Equation [21], so eddy currents will persist 13,700 times longer in an aluminum 10D. R. Lide, (Ed.), CRC Handbook of Chemistry and Physics, 72nd Edition, CRC Press, Boca Raton, 1991. structure at 10 K than one at 293 K. In a commercial aluminum alloy the conductivity will vary from that of the pure metal, especially at low temperature, so the effect may not be as great. In practice, the principal source of eddy current fields is generally the innermost low temperature aluminum cylinder, which is at approximately the boiling point of liquid nitrogen, 77 K. The resistivity of aluminum at 80 K is 2.45 x 109 Qm, so the time constant is about 11 times greater than it would be at room temperature. We consider the desirable characteristics for an eddy current measurement technique. Our primary goal will be to measure the eddy current field in order to evaluate the feasibility of performing a given experiment, not to compensate for the eddy current field. Therefore dynamic range is more important than absolute accuracy. It must be possible to measure eddy currents produced by specific pulse sequences, probably by appending the eddy current measurement experiment to the end of the sequence under evaluation. It is also preferable to have a technique that is insensitive to inhomogeneity so that no swimming is necessary. Since the shim coil power supply may respond dynamically to the gradient pulse and distort the measured eddy current field, it is useful to be able to turn the shim supply off. Experiments based on Selective Fourier Transform11 and other chemical shift imaging techniques rely 11H. R. Brooker et al., Maan. Reson. Med. 5, 417, 1987. for spatial localization on the integral of the eddy current field, so it is desirable to have a measurement technique that is based upon the integral of the eddy current field. If possible, the technique should have no special hardware requirements. Literature Review Many workers have addressed the problem of eddy current measurement and compensation in the literature. The two aspects of the eddy current field to measure are the spatial and time behaviors. We review publications that include descriptions of eddy current measurements, although in most cases the emphasis is placed upon the preemphasis compensation process and its effectiveness, not the measurement. The measurement process can be divided into techniques that detect the derivative of the eddy current field, those that detect the eddy current field itself, and those that detect the integral of the eddy current field. The derivative of the field is sensed by a pickup coil consisting of turns of wire through which the changing flux of the eddy current field produces an electromotive force that is proportional to the rate of change of the field.12 A high impedance preamplifier boosts the signal. An analog integrator is usually used to convert the measured voltage into a quantity proportional to the field, although it is possible to use digital integration. When used in a magnet 12D. J. Jensen et al., Med. Phys. 14, 859, 1987. at field, the pickup coil is sensitive also to any change in flux resulting from mechanical motion, which can contaminate the measurement. Since the field of the main magnet is generally about four orders of magnitude larger than the eddy current field and the time scale of mechanical modes is smaller than that of the longer timeconstant eddy currents, mechanical stability of the coil is crucial. Drift in the analog electronics is another potential difficulty with the pickup coil technique. Even with digital integration, the preamplifier can experience thermal drift on time scales not too different from the eddy current field. In spite of these difficulties, pickup coils are simple to use and can be used effectively to adjust preemphasis compensation. They are used routinely to correct for eddy currents in commercial, clinical MRI installations.13 A different approach to measuring the eddy current field is through its effect on the NMR resonance. One advantage here is that a pickup coil and its associated hardware are not needed. These proportional techniques measure a frequency shift in the NMR resonance that is directly related to the eddy current field.14 From Equation [13], the phase of freelyprecessing magnetization in the rotating frame at time t with respect to to can be written as 13Personal communication, Dye Jensen. 14Ch. Boesch et al., Magn. Reson. Med. 20, 268, 1991. 13 = yxJ gdt'. [22] The instantaneous frequency o(t), which can be defined as the rate of change of the phase of 0 by (0(t) = d)/dt is related to the eddy current field through the Larmor equation (0 = yB. Magnetic field homogeneity is important when using this approach, so that the FID will persist long enough to obtain a meaningful measurement. In another approach based on the NMR experiment, the phase of the magnetization 0 is measured at a single point in time. The phase at that point reflects the integral of the eddy current field over certain intervals in the experiment. Since only one point is sampled in each experiment, many more experiments are required to map the decay of the eddy current field than with the proportional techniques. However, T2* and offresonance effects do not affect the usefulness of the technique. The experiment proposed later is a singlepoint acquisition technique. All the techniques surveyed were implemented for unshielded gradient units, although preemphasis is typically used on systems with shielded gradient sets as well.15 Boesch, Gruetter and Martin of the University Children's Hospital in Zurich16,17 measure and correct eddy currents on a 2.35 T, 40 cm Bruker magnet. The unshielded gradient set has an inner diameter of 35 cm and a maximum gradient of 1 15R. Turner, Magn. Reson. Imaq. 11, 903, 1993. 16Ch. Boesch et al., Magn. Reson. Med. 20, 268, 1991. 17Ch. Boesch et al., SMRM 1989, 965. G/cm. They use two NMR techniques to measure the eddy current field. They interactively correct, using a 12 cm diameter glass sphere filled with distilled water, and they use no spatial discrimination in order to get all spatial components. The experiment consists of a 2.5 s gradient pulse of 0.6 G/cm followed by a train of 8 FIDs. There is a 20 ms delay between the time the gradient is switched off and the first radiofrequency (RF) pulse. The RF pulses have a 20 flip angle in order to reduce echo signals. The total eight FID acquisition time is 200 ms. They solve the Bloch equation for a sample with a single resonance frequency and decay constant and extract yABz(t) = (MydMx / dt MxdMy / dt) / (M2 + My) [23] as an estimate of timedependent Bo shift. They claim this gives enough information for interactive preemphasis adjustment. The one measurement they publish is of an already corrected system and shows 7ABz(t) decaying from 2 to 0 ppm as time t increases from 20 to 200 ms. Glitches are apparent at the ends of the FIDs. To map the spatial variations, they place a stimulated echo (STE) imaging experiment following the gradient pattern of the experiment they want to analyze. The STE sequence is applied with and without the preceding gradient pattern. The difference in phase is considered to be due to the time integral of the eddy current field in the interval between the first two pulses of the STE sequence. A series of slices tilted by multiples of 22.50 is obtained from the same 12 cm diameter phantom. The images were phase corrected. The phase of points along the z axis and on circles around the z axis was measured and used as data for a polynomial regression analysis to determine the coefficients of the various spatial harmonics. A table of the harmonic components following a 2.5 second x gradient pulse of 0.3 G/cm is presented. The delay between the end of the gradient pulse and the first RF pulse in the three pulse STE experiment is 20 ms, and the delay between the first and second RF pulse is 15 ms. The experiment was conducted following adjustment of the preemphasis unit. In decreasing order of magnitude, x, z, y, z2, xz2, xz, and x2 y2 terms were present. The value of the B0 term was not reported. Note that after x, the dominant terms should be eliminated by the symmetry of the coil/cylinder system. Only the x and xz terms would appear in an ideal system. The presence of terms having evenorder in x can be due to two reasons. First, the terms may really exist due to asymmetries in the magnet and gradient coil, crosstalk between amplifiers, etc. Second, the spherical harmonic analysis is highly sensitive to the point chosen to be the origin, and the most favorable origin may not have been employed. A series of the phasemodulated images is presented as well, with delays of 5, 20, 50, and 100 ms between the x gradient and the STE imaging sequence. The images are all from an already compensated system. Van Vaals and Bergman of Philips Research Laboratories in Eindhoven, the Netherlands,18,19 have a 6.3 T, 20 cm horizontal bore Oxford magnet with 2 G/cm non shielded gradients leaving a 13.5 cm clear bore. To measure the eddy currents, they use a 4 cm diameter spherical phantom. After swimming, they perform a simple "long gradient pulse, delay 8, RF pulse, acquire" sequence. The gradient is switched on for typically 3 s, but at least 5 times the largest eddy current time constant. For various values of 8, the magnet is reshimmed to maximize the signal during the first 10 ms of the FID. The difference in shim values with and without the gradient pulse is interpreted to be a spherical harmonic expansion of the eddy current field. Exact values of 8 are not listed, nor are tables of shim values. Instead, the amplitudes and time constants of the eddy current fields, as derived by a Laplace transform technique, are given. Only the B0 and linear terms are given; presumably only these terms were shimmed. Jehenson, Westphal and Schuff of the Service Hospitalier Frederic Joliot, Orsay, France, and Bruker,20 corrected eddy currents on a 3 T, 60 cm Bruker magnet. The 0.5 G/cm unshielded gradient coils had a clear bore of 50 18J. J. van Vaals and A. H. Bergman, J. Magn. Reson. 90, 52, 1990. 19J. J. van Vaals et al., SMRM 1989, 183. 20P. Jehenson et al., J. Macrn. Reson. 90, 264, 1990. cm. They use the same type of multiple FID sequence as Boesch, Gruetter and Martin, with an exponentially increasing sampling interval and 30 sampling points. The gradient prepulse is 10 s in length. The first FID is sampled at 1.5 ms after switching off the gradient, and sampling continues for 4 s using multiple FIDs. They plot the measured field vs. the time with and without compensation. They use a 1 mm by 3 mm waterfilled capillary positioned at +/ 5 cm to discriminate Bo and linear terms. They do not consider crosstalk or higher order terms. They use the same Laplace transform technique as van Vaals and Bergman, but they apply it iteratively to get better correction. Heinz Egloff at SISCO (Spectroscopy and Imaging Systems, Sunnyvale, CA)21 used a pickup coil to measure eddy current fields. To correct the B0 component of the eddy current fields, he moved the gradient coils until the field shift was eliminated. Riddle, Wilcott, Gibbs and Price22 considered the performance of a Siemens 1.5 T Magnetom. They measured the instantaneous frequency do/dt of a 100 ml round flask (presumably filled with water) following a 256 ms, 0.8 G/cm gradient pulse. They present plots for imaging and spectroscopy shims as well as for the gradient pulse. They endorse do/dt as an indication of shim. It would seem to 21H. Egloff, SMRM 1989, 969. 22W. R. Riddle et al., SMRM 1991, 453. work only for singleline samples, however. Following the gradient pulse, the plot of d'/dt contains peaks that are not explained. They may be an indication of the true do/dt, or they may be artifacts from beginnings and ends of FIDs. The sensitivity of the technique as presented here seems to be about 1 Hz. Hughes, Liu and Allen23 of the Departments of Physics and Applied Sciences in Medicine at the University of Alberta measured the eddy current fields of their 2.35 T, 40 cm bore Bruker magnet. After 57 delays ranging between 500 is and 2.5 s following a 0.2 G/cm gradient pulse the FID was measured and the offset frequency of the line determined. They placed a 13 mm diameter spherical water sample at +/ 1, 2, 4 cm along the axes of the radial gradients under test. A fourexponential fit was applied to all six locations simultaneously. The shortest time constant was associated with the amplifier rise time. An interesting plot shows that the field associated with each time constant is essentially linear. The Bo fields associated with the various time constants are different, however, suggesting a unique isocenter for each time constant. Zur, Stokar, and Morad24 of Elscint in Israel place a doped water sample at +/ 5 cm from the center in the direction of the gradient of the field. A train of 256 FIDs is acquired after switching off the gradient. Each FID is 23D. G. Hughes et al., SMRM 1992, 362. 24Y. Zur et al., SMRM 1992, 363. Fourier transformed, bandpass filtered, then inverse transformed. The instantaneous magnetic field is obtained from d)/dt. The digital filtering points to a problem with phase measurements. The lowpass filters required to eliminate Nyquist aliasing and to improve the signalto noise ratio (SNR) distort the phase of the received signal. Digital filtering enables one to recover the SNR ratio of a small bandwidth without significant phase distortion. Wysong and Lowe25 at Carnegie Mellon and the University of Pittsburgh measured eddy current fields on a Magnex 2.35 T 31 cm magnet with unshielded gradient coils. A 1 cm diameter sphere containing water doped to TT2~1 ms is used. A 0.9 G/cm gradient is applied for 1.0 s, then ramped down in 128 ps. A train of pulses of flip angle n/2 set 1 ms apart is applied for 1 second. One point is sampled for each FID. With the system adjusted so the FID is inphase in the absence of a gradient field, the outofphase component is proportional to sin(yABtet/2) = yABtet/2 for small values of time and gradients. Keen, Novak, Judson, Ellis, Vennart and Summers26 of the Department of Physics, University of Exeter, propose using a phantom slightly smaller than the imaging volume. Having switched off the gradient, they delay a variable time, then pulse and acquire the FID. The Fourier transform 25R. E. Wysong and I. J. Lowe, SMRM 1991, 712. 26M. Keen et al., SMRM 1992, 4029. of the FID represents a projection of the phantom in the quasisteady eddy current field. Measuring the distance between the peaks that appear as edge artifacts gives the eddy current field. Teodorescu, Badea, Herrick, and Huson27 at the Texas Accelerator Center and Baylor College of Medicine measured eddy current fields in their 4 T, 30 cm superferric self shielded magnet. The magnet was operated at 2.19 T. They follow Riddle et al.28 in their measurement. A small phantom is placed at various offcenter locations. They use a 0.8 G/cm gradient pulse of 15 ms and a 750 gs rise/fall time. This is followed by an FID (or a series of them) that is acquired for 20 ms. They compare this to the result obtained from a pickup coil. The eddy current field was measured with a sense coil and analog integrator by Morich, Lampman, Dannels, and Goldie.29 They used a Laplace transform approach to derive correct parameter values for an analog inverse filter to compensate for the eddy currents. The analog inverse filter was of conventional design,30 placed at the input of the gradient power supply. The theory was tested on an Oxford Magnet Technology whole body superconducting magnet. The approach is based on the ease with which a linear system can be analyzed in the reciprocal space s defined by 27M. R. Teodorescu et al., SMRM 1992, 364. 28W. R. Riddle et al., SMRM 1991, 453. 29M. A. Morich et al., IEEE Trans. Med. Imac. 7, 247, 1988. 30D. J. Jensen et al., Med. Phys. 14, 859, 1987. the Laplace transform. We can understand the calculation as follows. Assume the gradient field for t>0 in response to a unit step function is N g(t) = 1 aetw i i [24] i=1 The amplitudes ai and time constants Ti can be determined through a bestfit to experimental data. To determine the inverse filter, the first step is to deconvolve the step function to find the impulse response h(t), which can more conveniently be accomplished by a multiplication in the complex frequency space, s. The equivalent function G(s) is obtained by a Laplace transform 1 N a G(s) = _ N ai [25] S is + Wi Then the impulse response in the s domain, H(s), is found through the relation G(s) = H(s)/s, [26] so that N H(s) = sG(s) = 1 N ais [27] s + wi i=1 1 is the impulse response. The inverse filter's impulse response is just the reciprocal of the impulse response of the eddy currents, YH(s) = N [28] s + Wi The step response of the inverse filter, F(s), is the convolution of a step function and the impulse response: 1 1 F(s) = [29] sH(s) N ais2 S  S + wi i=1 i The amplitudes bi and time constants vi of the inverse filter can be read directly from the inverse Laplace transform, f(t), of F(s): N f(t) = 1 + biet/vi [30] i=l The inverse Laplace transform was performed by matrix inversion for a fourtimeconstant case using Gaussian elimination. Now the appropriateness of these techniques to the project of following the time evolution of the eddy current field can be considered. Two of the techniques, those of Egloff and Morich, involve the use of a pickup coil, preamplifier, and integrator. We choose to confine ourselves to NMR techniques. The procedures of Boesch, van Vaals, Jehenson, Riddle, Keen, Hughes and Teodorescu require swimming to correct for the inhomogeneity of Bo. The fact that T2* must be reasonably long also limits the region where eddy current fields can be measured to well inside the active imaging volume. Wysong and Zur propose similar NMR techniques that do not require swimming. In general, however, it is samples with long relaxation times that are most sensitive to small eddy current fields, and the use of a sample with especially short (T~T21 ms) relaxation times is not an obvious way to detect lowlevel fields. The T2 of the sample limits the duration of the interval in which phase can be sampled. Another drawback is that the trains of 7/2 pulses will produce stimulated echoes, even if T1 is on the order of the interpulse separation. However, this may be the most promising of the techniques surveyed. SpinEcho Techniques Distortions in the phase of spectra spatially localized with a twopulse Selective Fourier Transform technique31 were observed by Mareci.32 He observed that the distortions were reduced by lengthening the echo time, consistent with the known behavior of field distortions due to eddy currents induced in the metal structures of the magnet by the pulsed gradient fields used for spatial localization. We consider how a series of spin echo experiments identical except for n/2y it. RF nA g I I I 0 TE/2 TE Figure 1. Twopulse experiment with pulsed field gradient. The long trailing edge of the gradient pulse indicates distortion due to the eddy current field. 31H. R. Brooker et al., Magn. Reson. Med. 5, 417, 1987. 32T. H. Mareci, Personal communication. increasing echo time (TE) gives an indication of the eddy current field distortion as a function of time. Consider the evolution of the rotatingframe magnetization m in the presence of the gradient field g illustrated in the pulse sequence in Figure 1. For O Equation [13] so iyx f gdt' m(t) = m(O)e 0 < t < TE / 2. [31] We can also apply the result directly to describe the magnetization's evolution following the n pulse. Let TE/2+ be the time just after the n pulse. Then m(t) = m(TE / 2+)e TE / 2 5 t. [32] The t pulse along x inverts the sign of the imaginary part of m(t), equivalent to taking the complex conjugate: iyx JTE/2 gdt' m(TE / 2+) = m*(0)e 0 [33] Putting it together gives iyxITE/2 gdt' iE/2 gdt m(t) = m (0)e e TE / 2 < t [34] [ (TE/2 : , iyx gdt gdt' m(t) = m*(O)e J JTE/2 J TE / 2 < t. [35] Measurement of the phase exactly at the center of the Hahn echo should remove offresonance effects, whether due to chemical shift or field inhomogeneity. Now it remains to be shown that measurements of the phase at a series of echo times can be used to find g(t). If 00 is the phase without a gradient pulse applied, then (TE) 40 = yx E/2 gdt' J/ gdt] = yx2fTE/2 gdt' JTEgdt]. [36] We define a function G(t) by G(t) = yxJ gdt' [37] which simplifies the expression above for ): O(TE) 40 = yx[2G(TE / 2) G(TE)] [38] By measuring 00 and measuring ) at a number of echo times, we hope to be able to extrapolate the function G(TE), whose rate of change gives the eddy current field. By performing a series of experiments in which the values of TE are related by successive powers of two (TEi1j = 2TEi), we can obtain a series of coupled equations. Using the shorthand 0 (TEi) = 0j, Oi+1 = y [2G +i] i = 1, 2, [39] Inverting for G, yields GI = [(4i+l 0o)/yx + Gi+1]/2 i = 1, 2, ... [40] For large enough i, Gi = Gi+I, and the equation has an immediate solution. The remaining Gi can be determined recursively. The rate of change of G(TE) is the eddy current field. Experiments and subsequent data analysis have pointed to several drawbacks in this approach. The first is that the echo time TE limits the maximum length of the gradient pulse. A gradient pulse long in comparison to the eddy current decay time approximates a step function, which simplifies the analysis of the eddy current response.33 However, lengthening the TE reduces the time resolution of the experiment. Placing the gradient pulse before the excitation pulse as in Figure 2 eliminates the problem and decouples the length of the gradient pulse from the echo time. n/2y nx RFnA Id2 0 d3 TE/2 TE/2 Figure 2. Gradtest v.1.2 is a spinecho experiment for measuring the eddy current field following a pulsed field gradient. The above analysis assumes a point sample. Any real sample has finite extent and will experience some dephasing, and associated signal loss, as its phase evolves in the gradient field. By not subjecting the transverse magnetization to the gradient pulse but only to the eddy current field, the dephasing effect is reduced. Another drawback proved to be that the signal decayed due to T2 relaxation before Gi stabilized. With the gradient pulse before the excitation, the condition Gi = Gi, could be met for small values of TE. However, for large TE we could 33M. A. Morich et al., IEEE Trans. Med. Imaq. 7, 247, 1988. assume that g = 0 while for small TE, g # 0. To solve for the Gi it is necessary to know one of them in advance, so to determine Gi for large TE, another experiment was performed. TE was held fixed at a large value and d3, the interval between the end of the gradient pulse and the RF excitation pulse, was varied in steps of TE/2. A system of simultaneous equations describes the phase obtained by varying d3 in steps of TE/2, starting with d3 = 0: O(TE + d3) 00 = yx[2G(TE / 2 + d3) G(TE + d3) G(d3)] [41] The problem of signal decay due to T2 is thus circumvented. This technique could be used by itself or, as we used it, only to obtain a starting point for varying TE. A remaining difficulty is the ambiguity of phase measurement. Phase can be directly measured only modulo 3600, but the accumulated phase in our experiment may be much greater. One way around this difficulty is to reduce the applied gradient so that we can be sure that our sample rate is above the Nyquist limit, so that 4j+1 i < 1800 To get an upper bound that guarantees no phase ambiguity, assume that the eddy current field has the same amplitude as the applied field before the n pulse and zero amplitude following the 7 pulse. Protons process at 4258 Hz/G. To get a measurement for TE/2 = 512 ms without phase ambiguity would, for a sample 1 cm from the center, require a gradient pulse no greater than 0.000229 G/cm. Such a small gradient pulse would result in no detectable phase accumulation in practical cases. Experimental experience showed that it was not simple to choose in advance a gradient amplitude that would result in measurable phase accumulation, but no phase ambiguity, at all echo times. Instead, we repeated the experiment for a series of increasing gradient amplitudes. For phase changes of less than 3600, the phase doubles as the gradient doubles. We could keep track of phase accumulations greater than 3600, thereby decreasing the minimum detectable eddy current field. Stimulated Echo Techniques The stimulated echo (STE) has advantages over the spin echo as the basis of an eddy current field measurement experiment. Consider the stimulated echo sequence Gradtste in Figure 3. The "e" at the end of the pulse sequence name indicates that this is a stimulated echo experiment. A third pulse is required to excite a stimulated echo. The magnetization of interest is flipped into the transverse plane by the first RF pulse, where it accumulates phase RF n n I I I I I t T t grad decay 1 Figure 3. Diagram for Gradtste, a three pulse stimulated echo experiment for measurement of the eddy current field. shift due to static field inhomogeneity and eddy current fields. Then, stored by the second RF pulse along the z axis, the magnetization accumulates no more phase until the final RF pulse tips it back into the transverse plane. The phase accumulation due to static inhomogeneity now unwraps, resulting in the stimulated echo. If tj is long enough, there is essentially zero eddy current field in the second T, so the phase accumulated due to eddy current fields in the first T is preserved. It is possible to follow the eddy current decay by incrementing either tdecay or T between experiments. If T is incremented, the procedure for determining the eddy current field is similar to that for spin echo experiments. The phase shift for two experiments with different T is subtracted to get the integral of the eddy current field in the time between the earlier and later T. A more direct approach is to increment tdecay between experiments, keeping T small. Using this approach, each experiment yields the integral of the eddy current field over a short interval T. Dividing by T yields the average eddy current field in the interval. Two advantages of the STE are immediately evident. A single STE experiment can be directly related to phase accumulation in a single interval, eliminating the need for the recursive data analysis or simultaneous equations associated with the spin echo technique. This would also seem to make the choice of gradient pulse amplitude more straightforward. Since tI is limited by T1, which is generally longer than T2, it is possible to sample with smaller residual gradient field than in the spin echo experiment. The eddy current field is subject to a multiexponential decay. The integral of a multiexponential decay is another multiexponential decay. We can expect these functions to be reasonably smooth. That is, if we notice that the phase is not changing much between delay increments, we could either increase the delay increment or increase the amplitude of the gradient pulse. This is a form of adaptive sampling, since the sampling strategy for the gradient field depends upon its behavior. The sampling technique should be capable of following the residual field decay when preemphasis is used, and in this situation the field will not in general decay monotonically, since some of the decay components may be overcompensated. Therefore the adaptive sampling must also be able to decrease sensitivity when needed. Since the eddy current field generally changes most rapidly at short times, varying T to keep the measured phase shift approximately constant for each value of tdecay yields less densely spaced measurements when the field is changing slowly. We have implemented such an adaptive sampling technique by writing a recursive macro Adgrad in the Varian MAGICAL language to perform a series of measurements in which T is varied to "lock" the phase shift to 45. The macro functions as a command to the Varian program "VNMR" through which the spectrometer is controlled. Adgrad allows the automatic measurement of the eddy current field over a large dynamic range. Fortyfive degrees is large enough to measure with enough precision and yet small enough to minimize the possibility of aliasing. The values of phase 'user enters " Adgrad ( t max) " Figure 4. Flow chart of the macro Adgrad, which executes adaptive sampling of the eddy current field. The dotted portion is not part of the macro. shift A0 and T are easily reduced to a plot of the eddy current field vs. time. A flow chart of Adgrad is found in Figure 4. It is most easily explained in the context of the whole experimental procedure. The user notes the phase of the STE for an experiment with gph, the value of the gradient pulse, set to zero. He then selects a combination of T, tdecay, and gph that results in phase accumulation of about 450 and acquires an FID. He also removes the file "phase.out" if it remains from a previous session. Then he executes the macro Adgrad(tmax, o0), where tmax is the value of tdecay at which the macro will stop and 00 is the phase with gph = 0. Adgrad first calls the macro Calcphase to compute the phase A0 at the center of the acquisition window (that is also the center of the FID) for the data already in memory. Adgrad then stores the values of A0 and T as the first line in the output file "phase.out." Next, Adgrad tests to find if tdecay > tmax. If so, it ends the experiment. This should not occur on the first pass through the test. In the following two steps, Adgrad sets up the timing for the next experiment. The new tdecay is set to be greater than the old by T, to provide for a contiguous series of intervals t. The new T is set so that if the eddy current field remains constant, the next measurement will yield a phase A0 of 45. Now the measurement is started. Following the measurement, the macro calls itself and the process repeats. When the tdecay > tmax test is passed, Adgrad returns control to the operator. Two other adaptive sampling macros have been developed for eddy current testing. Adgrad2 changes both T and gph to lock the phase to 45. The resulting series of experiments are more closely spaced in time than Adgrad. Using Adgrad2, it is possible to follow the eddy current field over a wider range of values than with Adgrad. However, linearity error in the digitaltoanalog converter or nonlinear amplifier response will be reflected in error in the eddy current field. Adgradl80 locks the phase to 1800. It can only work when the phase accumulation is monotonically decreasing yet never changes sign, which is true for the uncompensated gradients. Otherwise, Adgradl80 may lose its lock. If the error in the measured angle is constant, the accuracy of the technique, when applicable, should be about 5 times better than for Adgrad or Adgrad2. In the preliminary data analysis, we assumed that the eddy current field was essentially constant over the sampling interval T, so that g = AO/yxT. An Excel spreadsheet was used to reduce and analyze the data and plot the results. An example is shown in Figure 5. It is a plot of the average field in each of the measurement intervals. Since the field is dropping exponentially, not linearly, fits to the mean value will have systematic errors. A better way is to assume a multiexponential decay of the eddy current field, and then calculate what phase will be measured in the STE experiment. If we define OI(t) as the total phase shift from t = 0 to tn for gph = 1, then n (D(t) = [42] i=l This phase shift is just, for a single experiment, *((t) = yx g(t'dt' [43] Now we assume that the eddy current field can be described by a threetimeconstant decay, g(t) = Ae tta + Bet/tb + Ce c. [44] Integration gives a function to which the measured phase can be fit: D (t) = yx[taA(l etta) + tbB(l et/tb) + tc( etc)]. [45] Results Eddy current measurements were made on several gradient coils of practical interest. Tests of the Oxford gradient coils in the 2 and 4.7 T magnets were conducted. For the 4.7 T magnet, the eddy current measurements were used to adjust the preemphasis network. Measurements of the eddy current fields associated with homebuilt gradient coils were also made. The detailed design and construction of the coils, on 9 and 15 cm former, is described in the following chapter. Initial measurements were made using the spinecho technique of Gradtest vl.2. A 5 mm NMR tube with about 5 mm of H20 trapped by a vortex plug was used as a sample and placed 1.7 cm from the center of a 2 T, 31 cm horizontal bore magnet (as measured from an image). The Oxford Z gradient in the Oxford 2 T magnet was pulsed to a value of 1000 units or 1 G/cm. The manufacturerinstalled preemphasis filter was in place. A d3 array with four elements was used to establish the phase value for large echo times via matrix inversion of simultaneous equations. An echo time array resulting in a series of coupled equations was used to work back to 1 ms. The resulting plot is shown in Figure 5. The bumpiness of the plot may be due to the preemphasis. Data points are plotted in the center of the interval for which they represent the average gradient. 0.045 0.04 0.035 0.03 U 0.025 0.02 m 0.015 0.01 0.005 0  I IIlli'  0 100 200 300 400 500 600 700 800 900 t (ms) Figure 5. Eddy current field as a fraction of applied field for Oxford gradient coil. Stimulated echo measurements using the pulse sequence Gradtste and the macro Adgrad were conducted for the Oxford gradients as well as for the 9 cm homebuilt gradient coil in the 2 T, 31 cm diameter magnet. The eddy currents for the Oxford gradients were measured with the manufacturer installed preemphasis filter in place. The 9 cm coils had no preemphasis. A 5 mm NMR tube with about 5 mm of H20 trapped by a vortex plug was used as a sample and placed between 1 and 2 cm from the center of the magnet. The center of the sample was determined from an image. In all cases, tgrad = 2 s, dl = 2 s, tj = 0.5 s, and two averages were acquired. The parameter T was set to 4 ms and tdecay was 1 ms for the initial experiment. The data were analyzed in Excel spreadsheets. In the plots of field vs. time given in Figures 6 and 7 for the Oxford and 9 cm coils respectively, the average of the eddy current field over the sampling interval is plotted against the middle of each sampling interval. The eddy current field is represented as a percentage of the applied gradient. Note that without preemphasis, the eddy current field due to the 9 cm coil declines monotonically, while the preemphasis filters contribute to the measured field of the Oxford coils. For the 15 cm coil tested in the 2.0 T magnet, eddy current measurement was used to calculate values for an inverse filter. The coils and samples were removed between the experiments before and after preemphasis. The Techron 7540 amplifiers were used to drive the coils in current mode. Hm r in (%) (^)6 _ , I I~ 11 I  C,, u H4 u 0 S L I (%) (4)B a) 00 r w in m (N H t 0 H I (%) (4)) S Al 0Co m in mm "m **. 4 *) 0 ,3 co 0)H H 0 H 0) rd d4 01 o * O u 44 0) H x eo 0 r r O 0 m 00 p H : 44 a) H *H (I (0) 0 rx.4 1 *  m U .m m * U * W .. (%) (q)6 r 0 0 0 0 (%) ( )B M * ,  I . (%) (1) r1 H r 0 (s u SH 0u u ( >4 EH H 0 0 A 0 U H o ro .,0 ) U2 O rd a, 0(U Oi  0(1) SH ., I >H H U rX : * N The same sample and RF coil were used as in the Oxford and 9 cm tests. In all cases, tgrad = 0.5 s, tL = 0.5 s, the sample position was between 1 and 2 cm from the center, and the position was measured in an image. The data were g(t) (%) 5 4 3 2 1 t (s) 0.1 0.2 0.3 0.4 0.5 g(t) (%) 6 4 3 2 1 t (s) 0.2 0.3 0.4 0.5 = t (s) 0.5 0.1 0.2 0.3 0.4 Figure 8. Eddy current field of 15 cm gradient coil set in 2.0 T magnet system before (upper curve) and after compensation (lower curve). a) X coil; b) Y coil; c) Z coil. g(t) (%) 6 5 4 3 2 1 0 acquired with Adgrad2, that changes gph as well as T to keep the phase locked. Eight averages were acquired. The eddy current field was measured out to 1 s, although the plots in Figure 8 only show 0.5 s. The data were analyzed both with the averagefield technique used in the Excel (Microsoft, Inc.) spreadsheet and with a multiexponential curve fit in Mathematica (Wolfram Research, Inc.). The curve fits seemed more satisfactory, and are shown in Figure 8. The lower curves represent the eddy current field after compensation. The curves plotted are the derivatives of the exponential curves that were fitted to the raw data. The preemphasis filter amplitudes and time constants were taken to be those of the eddy current field itself. This procedure should 0(t) (degrees) Echo Phase Shift 400 300 200 100 t (s) 0.2 0.4 0.6 0.8 1 Figure 9. Fit to raw data of eddy current field of Oxford Z gradient field for 4.7 T magnet system before compensation. tend to underestimate the preemphasis required, but since the unshielded eddy current fields were already less than 5% of the applied field, the error is not severe. The 4.7 T magnet that replaced the 2 T 31 cm magnet did not have manufacturerinstalled preemphasis, so the eddy current measurement techniques were applied to design an appropriate preemphasis filter. Since the uncompensated eddy currents were on the order of 50% of the applied field, the approximation used to compensate the 15 cm coil would not be effective. An inverse Laplace transform technique was used to design the filters. The technique was implemented through the symbolic inverse Laplace transform capability of Mathematica. An example of a multiexponential fit to the raw phase accumulation performed with Mathematica is shown in Figure 9. Eddy current fields before and after compensation are presented in Figure 10. The upper curves represent the field before, and the lower curves after, preemphasis. For the Y coil, the procedure was repeated a second time to obtain an additional reduction of the eddy current field. The lowest curve in Figure 10 (b) represents the eddy current field after the second pass of eddy current correction. Conclusion A technique to measure the eddy current field of a pulsed field gradient based on the phase of the stimulated echo NMR signal has been proposed. Experimental 42 g(t) (%) 50 40 30 20 0 t (s) a) 0.2 0.4 0.6 0.8 1 g(t) (%) 40 35 30 25 20 15 10 5 0 t (s) b) 0.2 0.4 0.6 0.8 1 g(t) (%) 40 30 20 10 0 t (s) C) 0.2 0.4 0.6 0.8 1 Figure 10. Eddy current field of Oxford gradient coil in 4.7 T magnet system before (upper curve) and after compensation (lower curve). a) X coil; b) Y coil. Lowest curve was acquired after secondpass preemphasis; c) Z coil. verification consists of measurements of the eddy current field before and after preemphasis. The level of the eddy current field after preemphasis can be interpreted as an upper limit on the error bar of the measurement. It is only an upper limit, since other errors also contribute to the residual eddy current field. Any error in the values of timing components in the preemphasis filter will add to the residual eddy current field. Also, any distortion in the amplifier will reduce the effectiveness of the compensation, since the filter was designed based on the assumption that the amplifier is linear. Error in the eddy current measurement technique itself might be due to other echo terms than the stimulated echo contributing to the signal. However, experiments have shown that other echoes are essentially negligible due to a combination of favorable timing and phase cycling. In the case of Adgrad2, which scales gph as well as T, it is clear that some error is due to inaccuracies in the digitaltoanalog converter (DAC) output level. The applied gradient is then not proportional to the DAC code, and so there is an error in normalizing to the applied gradient. Error in the curve fits may be significant, since in a multipleexponential fit it is difficult to get an accurate fit if the time constants are not widely separated. Note that secondpass adjustment of the preemphasis was more effective in reducing the residual eddy current field. The technique came out of a need to quantify phase distortions in localized spectroscopy. It is therefore bettersuited to measuring the time integral of the eddy current field than the field itself, and it is the integral of the field that gives rise to errors in phasesensitive techniques such as SFT. It is often useful to employ the basic stimulated echo experiment without adaptive sampling to quantify the integral of the eddy current field over an interval, and allow one to predict the resulting phase distortion directly. The adaptive sampling algorithm is able to follow eddy current fields that are not simply monotonically decreasing. I found experimentally that if the angle became much different than 450, the values for T would bounce around a lot before stabilizing. This is probably due to the control being purely proportional. Introducing an integral term might help. The relatively large eddy current field produced by the 15 cm Z gradient compared to the X and Y channels is due to its extendedlinearity design, which locates the currents farther from the region of interest than a Maxwell pair. The relatively large eddy current field produced by the 9 cm Y gradient compared to the X and Y channels may be due to a problem with centering the gradient coils in the bore. The measured field gradient would then depend strongly on the position of the sample.34 The contrast in eddy current field between the large and small coils is clear. There is a factor of about 10 in eddy current field between the 15 cm coil and the larger Oxford coil. There is a factor of about 180 in eddy current field between the 9 cm coil and the Oxford gradient coil set 34D. J. Jensen et al., Med. Phys. 14, 859, 1987. 45 in the X and Z channels. Experimental evidence demonstrates the advantage in eddy current field obtainable with reduced size gradient coils. GRADIENT COIL DESIGN Introduction and Theory Although virtually all NMR measurements rely on auxiliary field coils, there has been comparatively little published work on the design and analysis of shim and field gradient coils compared to that for radio frequency coils. However, high levels of performance have become increasingly important for these lowfrequency roomtemperature coils on several frontiers of the NMR technique. Three of these areas are gradient coils for NMR microscopy, coils for spatial localization of spectra, and local gradient coils for functional imaging of the human brain. The simple forms of discrete element coil designs have linear regions that are about 1/3 of the coil radius.35 Therefore the gradient coil must be considerably larger than the sample. Increasing the linear region would allow smaller coils to be used, generally improving efficiency and decreasing eddy current fields. Several approaches are available to increase the region of linearity. Adding discrete elements to cancel more highorder terms in the harmonic expansion has been done successfully by Suits and Wilken.36 Continuous current density coils have also been 35F. Romeo and D. I. Hoult, Magn. Reson. Med. 1, 44, 1984. 36B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989. designed with linear regions that are a large fraction of the radius.37 We have tried to take a fresh approach, combining aspects of both continuous and discrete designs. For a solenoidal main magnet, available radial gradient coil designs are longer and less efficient than axial designs, so we have chosen to concentrate on the radial case. z z e / 0 y X Figure 11. The coordinate system used in the text. An appropriate starting point to find a new radial gradient coil design might be: what current distribution on the surface of an infinitely long cylinder would produce a field in which the axial component is linearly proportional to the radial position, Bzo x? To describe surface currents and fields, we introduce the three coordinate systems described by Figure 11. Any point can be described in any of three orthogonal coordinate systems. In the Cartesian system a point is described by its location along the three axes (x, y, z). In the spherical system, it is described by two angles and the distance from the origin: 37R. Turner, J. Phvs. D: ADpp. Phys. 19, L147, 1986. (r, 0, )) In the cylindrical system, the point is described by (p, ), z). It can be easily shown that an azimuthal component of the surface current, J0, proportional to cos) and independent of z produces the desired spatial dependence. Neglecting for the moment the problem of current continuity, there are two possible approaches to achieving the cos) angular dependence. First, it can be approximated by superimposing azimuthal currents with no axial component. The solutions are exactly the same as for discrete filamentary currents. The first approximation, the 1200 arc familiar from the socalled Golay doublesaddle design,38,39 is shown in Figure 12(a). This class of designs has been called the "Golay Cage" because of its correspondence to the doublesaddle design. Higherorder approximations utilizing superimposed arcs are derived by Suits and Wilken.40 The other approach is to use our freedom to choose any axial current to meet J4o cos# by varying the current direction, for example, J4Oc cos#), Jzo sin). This approach leads to the Cosine Coil shown in Figure 12(b). Note that in Figure 12 the return paths are located away from the active volume of the coil. For a coil of practical length, the current return paths can significantly reduce and distort the gradient field. 38F. Romeo and D. I. Hoult, Magn. Reson. Med. 1, 44, 1984. 39M. J. E. Golay, Rev. Sci. Inst. 29, 313, 1958. 40B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989. x x (a) (b) Figure 12. Two radial gradient coils, a) The Golay Cage Coil; b) Cosine Coil. Another approach to current return paths is possible if we relax the requirement that the current is confined to the surfaces of cylinders. The current return paths can be located in the same plane as the azimuthal current paths. A gradient coil can be constructed of a stack of planes approximating a current sheet, such as shown in Figure 34 (a) on page 111. The planes include radial as well as azimuthal current elements. The radial currents do contribute to the axial magnetic field. It happens that the thirdorder harmonic terms eliminated by using 1200 arcs are independently zero for the radial currents connecting the arcs. These Concentric Return Path (CRP) Coils can have a linear region that can be increased in length by stacking more planes together. The overall combined coil structure can also be very short, since the return paths do not require extra length. In order to improve the linear region beyond that produced by a constant current density along z, we can adjust the relative current or position of each planar unit. Literature Review This literature review will be focused on efforts to increase the useful volume of a gradient coil, to optimize its performance, and to understand the eddy current field associated with a switched gradient it produces. The specific requirements of coils of interest for functional imaging of the human head are discussed, along with several approaches to meeting those requirements. Gradient coils can be grouped into two broad categories: those made up of discrete current elements as in Figure 13, and those approximating a continuous current density. The former include the original NMR shim coil designs,41'42 while the latter approach has been used to make possible actively shielded gradient coils.43 Anderson described a set of electrical current shims for an NMR system based on an electromagnet.44 The coils were located in two parallel planes, one against each poleface, to allow access to the sample. Each coil was designed to produce principally one term in the spherical harmonic expansion of the field. The orthogonality of the 41W. A. Anderson, Rev. Sci. Inst. 32, 241, 1961. 42M. J. E. Golay, Rev. Sci. Inst. 29, 313, 1958. 43P. Mansfield and B. Chapman, J. Maan. Reson. 66, 573, 1986. 44W. A. Anderson, Rev. Sci. Inst. 32: 241 1961. expansion ensured relatively independent adjustment of the current in the various coils. Techniques for designing higherorder shim coils for solenoidal magnets were set forth by Romeo and Hoult.45 Coils are designed by expanding the BiotSavart integral for Bz, the axial component of the field, in a spherical harmonic series about the center of the coil for simple filamentary building block currents. 1 Bz(r,9,()) = XAl,mPi(cos )ei. [46] 1=0m=1 The functions Pmf(cos ) are the associated Legendre functions. As building blocks are added in the form of arcs on the surface of a cylinder, more terms in a spherical harmonic expansion of the field can be set to zero. The designer connects the building blocks in such a way as to satisfy the requirement of current continuity, which is not built into the BiotSavart law. By setting each undesired term in the harmonic series to zero, a system of equations results. The solutions are the current, length, and position parameters of the coil designs. A Maxwell pair, as shown in Figure 13(b), is composed of a loop placed at 0 = 600 and another having opposite current direction placed at 8 = 120. This separation is required to cancel the (1, m) = (3, 0) term, while the odd symmetry cancels the (2, m) and (1, m # 0) terms. The desired (1,0) term remains. The 45F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984. simplest coil producing a gradient perpendicular to the axis of the cylinder is the double saddle or "Golay" coil illustrated in Figure 13(a). The arcs all subtend 1200 and are placed at the four angles 01 = 68.70, 02 = 21.30, 18001, and 18002, where they produce an (1, m) = (1, 0) term but no (1, m) = (3, 0) term. The relative current directions are shown in Figure 13(a). A family of solutions exists for which the sum of the (1, m) = (3, 0) terms produced by the arcs cancels, but the (1, m) = (3, 0) terms produced by each arc are not necessarily zero. We designate such coil designs by the two angles 01 and 02, so that the design above would be described as 68.70/21.30. Adding additional current elements to the coils adds degrees of freedom to the system of simultaneous equations, and makes it possible to cancel more terms. Adding another pair of loops adds two more degrees of freedom (the current and position of the new loops), and makes it possible to cancel higherorder terms including (5, 0). Note that the equations are not linear, so for a large number of current elements the procedure becomes unwieldy. For shim coils, it is less important to improve the linear region of a firstorder or gradient coil than to design additional coils whose lowestorder terms are of increasingly high order. The simple saddle coil in Figure 13(a) designed by this technique has a useful volume with a radius of about 1/3 that of the cylinder. x x (a) (b) Figure 13. Field gradient coils that use discrete filamentary current elements. a) Doublesaddle 68.70/21.30 coil to produce the radial field gradient x or y; b) Maxwell pair produces the axial, or z, field gradient. The approach is best suited to cases where the gradient coil is much larger than the sample, since the harmonic series approximation to the field converges more rapidly near the center of the coil. Although in theory the current elements are lines, in practice they do have finite dimensions, especially where large field intensity is required. Including the wire diameter would greatly complicate the design procedure. It is natural to use this technique to design the shim coils mentioned above, where it is conventional to have separate adjustments for as many as twelve or more terms in the harmonic series. The coils designed this way have the advantage of simplicity of construction. The building block approach was successfully extended by Suits and Wilken46 to use discrete wires to produce a constant field gradient over an extended region. They evaluated designs for cylinders with the polarizing field both parallel and perpendicular to the axis. To improve the useful volume of the radial gradient coil, they superimposed four saddle coils. The available degrees of freedom then included the number of turns in each of the four coils, the angular width of the four arcs, and the axial positions of the arcs. Systems of nonlinear equations result that were solved to null desired terms in an expansion of the field in orthogonal functions. Numerical plots demonstrate that the useful volume was extended to about eight times that of the simple saddle coil. In each case, the volume was nonspherical. The problems of extending this approach further are that larger systems of nonlinear equations are increasingly difficult to solve, and that the orthogonal expansions do not converge rapidly away from the center of the coil. Bangert and Mansfield47 designed a gradient coil in which the wires were included in two intersecting planes. The wires in each plane formed two trapezoids symmetrically placed about the z axis. By setting the angle between the planes to 450, the thirdorder terms in the magnetic field are canceled. 46B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989. 47V. Bangert and P. Mansfield, J. Phys. E: Sci. Instrum. 15, 235, 1982. Approaches based on a continuous current density are most often used for gradient coils, where performance and linear volume are more important than simplicity. The impetus for these coils was the echoplanar imaging technique of Mansfield,48 which requires high intensity field gradients switched about an order of magnitude faster than conventional Fourier imaging. Also, shielded coils are useful in other imaging experiments that require rapidly switched gradient fields, and in volume localized spectroscopy. Without shielding to cancel the external field, the higher frequency and intensity lead to greater eddy currents in the cryostat and magnet, that in turn distort the linearity and time response of the field. Using current on two concentric cylinders, it is possible to produce a linear field inside the inner cylinder and zero field outside the outer cylinder. Continuous current density coils can be designed to have a large linear region, and, since current flows on the surface of the whole cylinder, high efficiency. A Fourier transform technique was applied by Turner to design gradient coils that approximate a continuous current distribution. The approach arose from consideration of the eddy currents induced on cylindrical shields concentric to gradient coils made up of discrete arcs.49 Expansion of the Green's function in cylindrical coordinates was a natural 48p. Mansfield and I. L. Pykett, J. Macn. Reson. 29, 355, 1978. 49R. Turner and R. M. Bowley, J. Phys. E: Sci. Instrum. 19, 876, 1986. approach to calculating the eddy current distribution. It was then possible to write the field produced by a general current distribution on the surface of a cylinder as a FourierBessel series.50 An inverse Fourier transform of the FourierBessel series allowed the current to be expressed in terms of the desired field on the surface of an imaginary cylinder. The field must satisfy Laplace's equation to allow the existence of the inverse Fourier transform. So by specifying the inverse Fourier transform of the desired field, the current distribution required to generate that field could be calculated. The continuous distribution of current is approximated by discrete wires. The wires are placed along the contour lines of integrated current. Although the principal application of the technique was shielded gradients, unshielded coils having extended linearity were also designed. For example, a radial gradient coil is reported to have a gradient uniform to within 5% over 80% of the radius and a length of twice the radius. The overall length of the coil is about 9 times the radius. It was pointed out by Engelsberg et al. for the case of a uniform solenoid that the homogeneity of the coil depends strongly on the radius of the target cylinder.51 They note that the field has the target value only on the surface of the target cylinder. For example, in order to achieve a 50R. Turner, J. Phvs. D: Appl. Phys. 19, L147, 1986. 51M. Engelsberg et al., J. Phys. D. 21, 1062, 1988. homogeneous field along the axis of the solenoid, the target cylinder should be as narrow as possible. The effect is especially pronounced at the ends of the target cylinder. The importance of functional imaging of the human brain and its reliance on the echo planar imaging technique puts special demands on the rise time and field of the gradient coil. The fact that smaller coils will be more efficient and less affected by eddy currents has motivated several workers to design gradient coils that will fit closely over the head. To use a small gradient coil it is necessary to have extended linearity in the radial and axial directions. For a head coil, extended axial linearity is especially important to allow the diameter of the coil to be smaller than the width of the shoulders. Wong applied conjugate gradient descent optimization to the design of gradient coils with extended linearity.52 He allows the position of current elements to vary to minimize an error function. It is possible to define the error function as desired, so it is simple to optimize over regions of any shape, or for coil former of any shape. It is also simple to include parameters such as coil length. Repeated numerical evaluation of the BiotSavart law for the test wire positions would limit the application to coils with a fairly small number of elements. Wong applied the technique to the design of a local gradient coil for the 52E. C. Wong et al., Maan. Reson. Med. 21, 39, 1991. human head.53 Its overall length was 37 cm, diameter 30 cm. The region of interest is a cylinder 18.75 cm in diameter and 16.5 cm long, over which the RMS (root mean square) error in the field was less than 3% for all three axes. The gradient coil was symmetric to avoid torque. In order to make still shorter coils, Wong placed the return paths on a larger cylinder.54 The wires on the inner cylinder were connected to the return paths on the outer cylinder over both endcaps. A coil was designed of 30 cm length, 30 cm inner diameter, and 50 cm outer diameter. The optimization region was a cylinder 24 cm long and 20 cm in diameter, and the RMS error over the cylinder was 7.2%. The symmetry of the coil eliminated the torque that arises in other short designs. Additional points on a cylinder 70 cm in diameter were added to the region of interest to force some partial shielding. Another approach to the design of gradient coils that will fit over the head is to design a coil that has its linear region at one end. Myers and Roemer55 used only half of a conventional coil to move the linear region to the end. A target field approach was used by Petropoulos et al. to design an asymmetric coil with low stored energy.56 The coil simulated was 60 cm long and 36.4 cm in diameter. The "center" of the coil was 14.5 cm from one end. The stored 53E. C. Wong et al., SMRM 1992, 105. 54E. C. Wong and J. S. Hyde, SMRM 1992, 583. 55C. C. Myers and P. B. Roemer, SMRM 1991, 711. 56L. S. Petropoulos et al., SMRM 1992, 4032. energy for a gradient of 4 G/cm was calculated to be 7.93 J. Since these coils can be made much smaller than the bore of the magnet, eddy currents are not a serious problem and neither of these coils is shielded. Unlike symmetric designs, these coils experience a net torque in the magnetic field that is potentially dangerous. Another coil at a larger radius can be used to cancel the torque experienced by an asymmetric gradient coil. Petropoulos et al.57 took this approach to design a head coil with an inner diameter of 36.4 cm, the same as their singlelayer coil described above, and an outer diameter of 48 cm. The length of both inner and outer coils was 60 cm. The coil was designed to have a useful region that is a sphere of 25 cm diameter. There is a price to pay in increased stored energy, which increases over the single layer coil value of 7.93 J to 19.2 J. Torquecompensating windings can be added to the same cylinder as the primary coil, resulting in a long structure one end of which is placed over the head of the patient. Abduljalil et al.58 developed such a coil set for echoplanar imaging. The diameter of the two radial coils was 27.2 cm and 31.2 cm. The center of the linear region was 17 cm from the end. The overall length was not reported, but based on artwork for the wire pattern, it seems to be about 116 cm. 57L. S. Petropoulos et al., SMRM 1993 1305. 58A. M. Abduljalil et al., SMRM 1993, 1306. Turner has suggested that the best approach to a compact gradient head coil design is that of Wong, in which the return paths are placed on a larger cylinder.59 He points to the trapezoidal gradient coil designed by Bangert and Mansfield,60 and discussed above, as a starting point for this approach. The concept for such a gradient coil is described in a patent by Frese, for a cylindrical geometry.61 It can be thought of as a Bangert and Mansfield coil in which the inner and outer wires have been stretched into arcs on concentric cylinders. This is the design independently developed by Brey and Andrew and dubbed the Concentric Return Path Coil (CRPC). Frese suggested using a stack of the planar CRPC units with spacing along the cylinder's axis varied to improve size of the linear region. He also suggested that the angle of the arcs could be varied from plane to plane. No specific information on the spacing or angle of the arcs is provided. A survey of the literature suggests that it is desirable to design a short gradient coil using the basic concentric return path geometry to be used for the human head. A direct errorminimization technique is appropriate for two reasons. First, the FourierBessel transform technique, although computationally efficient, limits the shape of the region of optimization to the surface of a 59R. Turner, Maan. Reson. Imaa. 11, 903, 1993. 60V. Bangert and P. Mansfield, J. Phys. E.: Sci. Instrum. 15, 235, 1982. 61G. Frese and E. Stetter, U. S. Patent 5,198,769, 1993. cylinder, and axial linearity is important for the head coil. Second, the currents are not confined to the surface of a cylinder, and the transform technique in its present form allows only for current on the surface of a cylinder. Field Linearity An extended linear region is one of the goals of a reducedsize gradient coil. In order to evaluate a coil design in terms of its linear region, it is necessary to define the boundary of the linear region. An appropriate definition for the error associated with a field gradient reflects the purpose of the gradient coil. In an error minimization technique, the error definition is central to the coil design. A reasonable parameter to use is the error in the field, B.E.= Bz() where the desired gradient, G, is G x measured at the center of the coil. Another error parameter is the error in the gradient, G.E., defined by 1 dBz(x) G.E.= In an NMR image, error due to the IGI dx gradient coil simply produces an error in the mapping between the sample and the image. The absolute mapping error is simply the error in the field, B.E. In practice, samples are usually centered in the gradient coil, so we may want to weight the error toward the center of the coil. We use an error parameter that corresponds to the mapping shift relative to the component of the distance to the center in the direction of the gradient, the relative error B (x) G x R.E., defined by R.E.= Bz) G G x Efficiency In order to make use of the efficiency the reduced size of an extendedlinearity gradient coil design can provide, it is necessary to construct the coil in such a way that it can be driven efficiently by an amplifier. By adjusting the number of turns, it is possible to trade maximum gradient for switching time. We will show that to obtain optimal switching time, the amplifier should be currentcontrolled to compensate for the inductance of the coil. To reduce switching time with such an amplifier, the coil resistance per turn should be as low as possible, even though the time constant of the coil will be lengthened. A timedomain model for the coil and amplifier will be used to explore the tradeoff between maximum gradient and switching time. L c R, Figure 14. Timedependent voltage source v(t) drives inductive load. We show that a currentcontrolled amplifier gives better switching performance into an inductive load than a voltage source. The amplifier, modeled by a timedependent voltage source, v(t), is connected to a load with resistance, Rc, and inductance, Lc, as shown in Figure 14. When a demand is applied to a currentcontrolled amplifier for some current, io, it will by definition change its output voltage, v(t), as much and as rapidly as possible to change the current through an inductance across the output. If the maximum output voltage of the amplifier is Vo, and we define the steady state output voltage v0 = Vo / Rc, where Vo>v>O, then the amplifier output voltage and current as a function of time will be 0 t < 0 0 t < 0 v(t) Vo 0 < t < to i(t) = I. 1 et/ 0 < t < to Vo to < t < t Rc i to < t< [47] where to is the time at which the output current reaches the desired current io, and T = Lc/Rc. It is straightforward to calculate that to = TIn Vo [48] The smaller the ratio of V0 to vo, the greater the switching time to will become. If the amplifier is a voltage source, the desired current will never be exactly reached. It is more desirable to use a currentcontrolled amplifier for which o>>v0. It will be shown that, with a currentcontrolled amplifier, additional series resistance per turn always decreases performance. Therefore, the series resistance should be reduced as much as possible, for example, by using a larger crosssectional area for the winding in a coil of fixed radius. Consider again the time response of a currentcontrolled amplifier from Equation [47]. Any internal amplifier resistance can be included in Rc to avoid any loss of generality. Assume there is some finite, positive Rc that maximizes i(t). Then for that di R, = 0. Solving for Rc: dRc di V + Vo [491 di =_ ie L+ e = 0 [49] dRc Rc2 c c L and assuming that Rc and Vo do not vanish, 1 e + R/c L e = 0. [50] This can be written as (1 + x)ex = 1 where t/T = x, [51] therefore ex = + x. [52] There is no positive value of x that satisfies Equation di [52]. Hence < 0 for all t>O, Rc>O. A lower Rc is dRc always an advantage when using a currentcontrolled amplifier, although the time constant T = Lc/Rc of the gradient system increases as Rc decreases. It is then appropriate to maximize the crosssectional area of the windings subject to considerations of linearity and available space. Next we consider how the number of turns of wire in a coil of fixed crosssectional area can be varied to achieve the desired performance. It is important to note that the time constant of a coil, for a fixed area, can properly be considered to be independent of the number of turns. This result follows from consideration of a gradient coil at low frequencies, as described by the equivalent circuit of a series resistor Rc and inductor Lc as shown in Figure 15. LC Re Figure 15. Equivalent circuit of a gradient coil in the lowfrequency limit. Let R1 be the resistance, and let L1 be the inductance of a single turn coil. It is well known that the inductance of a coil increases as the square of the number of turns of wire N.62 The resistance also increases as the square of the number of turns if the area is held constant, since as the number of turns increases, the area of each turn diminishes. 62T. N. Trick, Introduction to Circuit Analysis, p. 256, John Wiley and Sons, New York, 1977. The total resistance and inductance are then Rc = R1N2 Lc = L1N2. [53] The time constant T of the coil is just the ratio S= Lc/ R [54] Perhaps surprisingly, T is independent of N. The result does not apply when additional turns of wire are added to an existing coil, thus increasing the area. However, since the area should already be as large as possible to maximize the performance, it will not be possible to increase N without decreasing the size of the wire. To determine how many turns of wire N should be used in the gradient coil, we consider how rapidly and to what value the current rises for various N, holding the area constant. It will be shown that with a currentcontrolled amplifier, the coil is optimized to switch to the field at which it reaches a saturation current, IO, which is the maximum that the amplifier can supply. Consider an amplifier with negligible output impedance switching at time t = 0 from zero current to maximum current, I0, through a gradient coil, reaching I0 at to. Assume that Rc < VO/IO. The current as a function of time is: 1e1 0 t 5 to i(t) = Rc [55] TO t > to We define a current efficiency k for a single turn so that the gradient field G(t) = kNi(t). Rc varies as N2, and the magnetic field G varies as N, so kVo / f Wo1 e 0 < t < to G(t) = RIN [56] kNIo t 2 to A plot of G(t) for various values of N is shown in Figure 16. All three curves have the same time constant, so G(t) 0.175 0.15 0.125 0.1 0.075 0.05 0.025 0.1 0.2 0.3 Figure 16. Magnetic field produced by a current controlled linear amplifier coupled to a coil of fixed dimensions. Each curve represents a different number of turns. the difference in slope is due to the relative amplitude of the maximum gradient. The dotted line connects all the currentlimit points. Since at the currentlimit point to the amplifier is both a voltage and a current source, we can eliminate G(to) in favor of N and to, yielding an optimum number of turns for a given switching time. Nopt(to) = [1[ e [57 By substituting [57] back into [56], we obtain an expression for Gmax(to), the maximum field attainable at a given switching time for a class of coils having the same design except for the number of turns. Gmax(t0) = k I 1 et [58] YR, Gmax(to) is just the dotted line in Figure 15. The tradeoff between switching time and field strength is described by the plot of Gmax(to). In summary, a design procedure has been developed for optimizing the switching performance of a gradient coil. Use of a currentcontrolled amplifier reduces switching time. The crosssectional area of the winding is maximized subject to constraints that include linearity and available space. Then the number of turns is computed from Equation [57], given the desired switching time to. The resulting coil will give the largest possible gradient for the desired switching time. Eddy Currents Shielding efficiency of selfshielded gradient coils is typically evaluated using a screening factor, a ratio of the magnetic field outside the unshielded coil to the field outside the shielded coil.63 It is possible to take this 63R. Turner, Maan. Reson. Imaa. 11, 903, 1993. type of approach further and evaluate the ratio of the gradient at the center of the coil with and without the shield. This would seem to be a useful approach when evaluating reducedsize gradient coils and comparing them to shielded coils. For small eddy current fields, as in the case of reducedsize coils, an iterative approximation technique described below can be used to solve the integral equation for the eddy current field. This technique is best suited to situations where the eddy current field is much smaller than applied field, so that a firstorder approximation can be used. However, it is simple and flexible. To estimate the eddy current field due to a gradient coil, we assume that there is a passive shield surrounding the coil. The shield is typically part of the cryostat. The boundary conditions at the shield will be (B2 B1) n = 0 [59] SX (H2 H1) = K [60] where B1 and H1 are the magnetic induction and field inside the shield, B2 and H2 outside the shield, K is the surface current on the shield, and n is an outwardly directed unit vector normal to the surface of the shield.64 We assume the shield is perfectly conducting, so that with H2 = 0, H1 x n = K, [61] 64J. D. Jackson, Classical Electrodynamics, p. 1.5, John Wiley & Sons, New York, 1975. or more conveniently, 1 K = B x p [62] go since B = o0H and n = p at the cylinder. Recall the BiotSavart law: B(x) = J(x') x d3x'. [63] 4r Ix x'13 Let BO(x) be the freespace field from the gradient coil. Then B(x) = B0(x) + K(x') X dx [64] 4 x x'13 and substituting Equation [62] for the surface current, for 9 = Io, 1 X X' B(x) = Bo(x) +  B(x') x p'] x d x [65] 4 ix x'3 where p' = p(x'). This is an integral equation for B. We can solve it iteratively. If we define Bn(x) as the field to nth order, then the firstorder solution is B1(x) = Bo(x) + f[Bo(x') x p'] x d x. [66] 4x Ix x'3 The firstorder solution does not take into account eddy currents induced by eddy currents. When the coil and shield are in close proximity, not only are the eddy currents larger but they are also closer to the coil, so the secondorder effect can be important. To secondorder, 1 x X d2, B2(x) = BO(x) + [Bl(x) x '] d2x. [67] 4t Ix X'1 The expressions to first and secondorder for the eddy current field will be used to evaluate numerically the eddy current field of several coil designs. Although the result is not exact, the expressions can easily be integrated for coils and shields of totally arbitrary shape, assuming they are not too close together. The firstorder calculated eddy current field of a 68.70/21.30 radial gradient coil is plotted in Figure 17. The firstorder approximation breaks down for ratios of shieldtocoil radius of less than about 1.5. shield radius / coil radius 1.5 2 2. 3.5 4 20 40 60 eddy current field (%) Figure 17. Eddy current field of 68.70/21.30 doublesaddle radial gradient coil. The field as a percentage of applied field is plotted against the ratio of shield radius to coil radius. Coil Projects Coil projects were intended to meet experimental needs while exploring some aspect of coil design. The 15 cm, 9 cm and 16 mm NMR microscopy coils are well separated from any sources of eddy currents, and demonstrate the results that can be achieved with simple filamentary designs and without shielding. The CRP coil development was begun to produce a coil with good axial linearity for NMR microscopy, so that long, narrow samples could be observed. It seemed to be wellsuited for use as a head coil for echo planar imaging, and we turned the development toward that possible application. Amplifiers Three Techron 7540 dualchannel amplifier units (Crown International, Elkhorn, Illinois) are used to drive the threeaxis gradient coil sets. Each axis of the gradient coil set is split into two halves, and one channel of each amplifier unit is wired to each half. The plane in which the field is always zero can be shifted slightly by varying the relative gain in the amplifiers. This is particularly useful in the 51 mm, 7 T magnet, since the sample is inaccessible once loaded, and mechanical centering is difficult. The amplifiers are rated to produce 23.8 A at 42 V direct current output. The maximum slew rate is 16 V/gs. The output impedance is less than 7 mnQ in series with less than 3 gH, which is negligible. The power response into a 4 73 0 load is +/ 1 dB up to 25 kHz for 265 W. The noise is rated to be 112 dB below the maximum output from 20 Hz to 20 kHz.65 Tests of the Techron 7540 were conducted into six loads consisting of wirewound resistors between 1 and 9 Q. The amplifiers were pulsed to saturation at low duty cycle. 1090% rise times were between 4 and 6.5 9s, and so are essentially independent of load. Thus the amplifier was bandwidth limited, not slewrate limited, and it is appropriate to use a linear model. The voltage and current 60 16 T 50 14 12 40 1 4 U 10 o 30 U 8 20 6 4 > H 4 10 2 0 I I 0 0 2 4 6 8 10 0 2 4 6 8 10 R (ohms) R (ohms) (a) (b) Figure 18. Output of Techron 7540 measured into load. a) Measured voltage; b) Calculated current. produced are shown in Figure 18. For load resistance of four ohms or more, the amplifier at saturation can be modeled by a 56 V voltage source. For a load resistance of 65Crown International, Techron 7540, Elkhorn, Illinois. four ohms or less, it can be modeled as a 15 A current source. The Techron amplifiers are equipped with optional currentcontrol modules. With current control switched on, an amplifier behaves like a voltagecontrolled current source. With current control switched off, an amplifier behaves like a voltagecontrolled voltage source. Current control serves two functions when driving gradient coils. It compensates for any variation in temperature of the gradient coil due to resistive heating. More importantly, it enables the coil to be switched to low fields much more rapidly than the coil's time constant would otherwise allow. The currentcontrol module compares the demand (or input signal) to the voltage across a small shunt resistor in the output circuit. With a highly inductive load such as a gradient coil, at high frequencies the amplifier's output voltage is shifted almost n/2 with respect to the coil current, and the amplifier is unstable and will oscillate. The voltage and current response of one of the Techron amplifiers in current mode is shown in Figure 19. The controlled voltage overshoot reduces the current switching time. Approximately 5 A is being switched into a 7 Q load. An adjustable resistorcapacitor (RC) network in parallel with the coil rolls off the high frequency gain to compensate for the instability. The values of the RC network are determined by the inductance of the coil. Since the 7540 amplifiers are used with more than one coil, the currentcontrol units were modified so the RC networks can be plugged in and out when gradient coils are changed. time 1.79 ms Figure 19. Output voltage and current of Techron 7540 amplifier with currentcontrol module. The load is the highly inductive 9 cm field gradient coil. The amplifier rack was equipped with wheels and shared between the NMR microscopy and smallanimal spectrometers. It was used in voltagecontrol mode with the NMR microscopy system, and currentcontrol mode with the small animal system where the coil inductance was much higher. Input and output connectors were standardized to facilitate quick conversion. A fullyshielded output cable terminated in a fuseandfilter chassis eliminated interference from the RF coils. 16 mm Coil for NMR Microscopy The 16 mm gradient coil was developed as part of the NMR microscope development project described below. Earlier NMR microscopy gradient coils described in the literature were located outside of the RF probe insert, as part of the shim coil set. A simple and straightforward approach to improving the coil switching time, increasing the field strength, and decreasing the eddy current field is to integrate the gradient coils with the RF probe. This also allows the use of a narrowbore (51 mm) magnet. Drawbacks to this approach include a lack of flexibility. If the gradient coil is outside the RF probe, then any RF probe can be used. In our approach, one gradient coil is required for each RF probe. Also, since one of the dewars associated with the variabletemperature (VT) control system is replaced by an acrylic tube, the range of the VT system is reduced. Our probe did not contain any VT control capability. The fact that the coil former was so small encouraged us to choose a simple design to ease the assembly. Since the sampletube inner diameter was 4.5 mm and the first metal tube, or shield, had an inner diameter of 33 mm, this was a favorable case for using a reducedsize gradient coil. A simple 68.70/21.30 radial gradient coil as described above has a useful volume with a diameter of about 1/3 that of the coil,66 so the gradient former was chosen to have a diameter of about 15 mm, or 5/8". A factor of two remains in the ratio of the coil to the shield diameter. This results in an eddy current field for the 68.70/21.30 Golay radial gradient coil, based on Figure 17, of about 20% of the applied field. The NMR microscope gradient coil set is of the conventional Maxwell and Golay design described above. It was constructed to accommodate standard 5 mm tubes used in analytical NMR work. The 10 turns of 36 AWG enameled magnet wire are wound on a 5/8" nominal outer diameter acrylic tube (15.9 mm). Using a value of 1.36 Q/m for the wire67 and a length of 0.135 m per turn, the resistance of each side of the coil is 1.84 Q. The coil inductance can be estimated68 to be about 8 LH. The time constant of the coil is then about 4 Js. The current efficiency of a 68.70/21.30 Golay radial gradient coil is 0.918/a2 G/cmA, where a is the coil radius,69 so the coil has a current efficiency of 14.1 G/cm A. A Maxwell pair has a current efficiency of 0.808/a2 G/cmA, so the coil has a current efficiency of 15.3 G/cmA. The typical figure for the linear region of 1/3 the diameter of the coil is then enough to accommodate a sample. The coil is driven by the Techron 7540 amplifier set. The coil 66F. Romeo and D. I. Hoult, Maan. Reson. Med. 1, 44, 1984. 67D. Lide,(Ed.), CRC Handbook of Chemistry and Physics, 51st Edition, CRC Press, Boca Raton, 1970, p. 1529. 68F. E. Terman, Radio Engineers' Handbook, McGrawHill, New York, 1943. 69F. Romeo and D.I. Hoult, Maan. Reson. Med. 1, 44, 1984. has a small time constant, so using the Techron in voltage mode does not limit the switching time. The two halves of each gradient coil are driven separately. Since the voltage gain of the amplifiers can be adjusted manually, it is convenient to vary the relative gain in the coils to shift the zero point of the magnetic field to make up for sample misregistration. The details of the coil construction are visible in Figure 47 in the following chapter. The radial coils were wound on a flat winding former, then removed and attached to the acrylic tube with epoxy. To eliminate any solder connectors within the coil, the winding former allowed two loops to be wound at once, held apart at the correct distance. General Electric #7031 varnish was used to hold the wires together while the coil was being clamped to the former. No attempt was made to arrange the wire into a packed structure. The Maxwell pair was wound around the radial coils. The whole assembly was potted in epoxy to secure the coils to the former, and the 36 AWG wires were run down to a small printed circuit board mounted to the structure of the probe. It was necessary to pot the fine wires to keep them from moving in the magnetic field when a current pulse is applied. An example of the results obtained with the coil is reproduced in Figure 60. Although the coil is capable of about 150 G/cm, in routine operation, the coil was operated at a fullscale field gradient of 5 G/cm for the radial gradients, and 10 G/cm for the axial gradient, to allow sufficient resolution. 9 cm Coil for Small Animals An NMR magnet is frequently used for samples or animals significantly smaller than the available bore size. It is possible to take advantage of this fact and scale the size of the gradient coil to match the size of the sample. One advantage that accrues is reduced eddy current fields, since the coil and the source of eddy currents are better separated. Another is the increased efficiency possible with smaller coils, since efficiency scales as the fifth power of the diameter.70 Many applications require more rapidly switched and more intense gradient fields than are generally available. Diffusionweighted imaging and localized spectroscopy are two examples. Also, to achieve the same bandwidth per pixel, small samples require larger gradient fields. The 31 cm 2 T smallanimal imaging spectrometer was supplied with a gradient coil set manufactured by Oxford instruments that has a clear bore of 22.5 cm, and is capable of producing a maximum gradient of 2 G/cm with a switching time of 1 ms. Although rat, mouse, and lizard studies, do not require the full 22.5 cm bore, they benefit from the horizontal orientation and will not fit into other available magnets. Additionally, localization techniques such as 70R. Turner, Macn. Reson. Imaq. 11, 903, 1993. selective Fourier transform71 typically require better gradient performance than is available with a large, unshielded gradient coil set. To meet some of these needs, a conventional Maxwell and 68.70/21.30 Golay radial gradient coil set was designed and constructed with a clear bore of 8.3 cm in diameter. The useful region is a sphere of about 1/3 the diameter of the coil, or about 3 cm. The coil was designed to accommodate rats up to 150 g, and was able to achieve 12 G/cm with a 200 ps switching time. The intense gradients are needed for imaging experiments on smaller samples. The coil was used for projects involving lizards, and for development of techniques to produce diffusion images of the spinal cord of a rat model. We can consider the application of the timedomain model to the 9 cm coil. As wound, the coil will produce the field shown in Figure 20 when driven to saturation. The Maxwell pair, Zaxis coil, is the most efficient, followed by the inner radial, or Xaxis coil, which has better performance than the outer radial, or Yaxis coil, because of its smaller radius. The Maxwell pair reaches the 16 A current limit of the amplifier, and does not increase in field beyond that point. The radial gradient coils never reach the current limit. 71T. H. Mareci and H. R. Brooker, J. Maan. Reson. 57, 157, 1984. G(t) (G/cm) f t (ms) 0.2 0.4 0.6 0.8 1 Figure 20. The gradient produced by the 9 cm gradient coil set following a demand that saturates the amplifier. Figure 21 describes the maximum field Gmax(to) possible for switching time to for each of the three coils. The Gx (G/cm) 25 20 15 10 5 I .. .' to (m s ) 0.2 0.4 0.6 0.8 1 Figure 21. The maximum gradient that could be achieved by a coil identical to the 9 cm coil, with the same cross sectional area, but with varying number of turns. inherently lower inductance and resistance of the Maxwell pair are reflected in its greater field. The actual and optimal fields obtainable at 200 ps are compared in Table 1. The Maxwell pair has about the optimal number of turns, and its field is about the same as the optimum. The Golay coils have about twice as many turns as optimum, and yield fields about 80% of optimum level. Table 1. Gradient fields for 9 cm coil set. Gradient Actual Gradient Optimal Gradient channel no. of after 200 no. of after 200 turns ps (G/cm) turns Rs (G/cm) X 52 15.2 27.9 19.0 Y 52 12.3 26.9 15.9 Z 52 24.0 53.6 24.8 The radial and axial gradient coils consist of 52 turns of AWG 27 enameled magnet wire. The wire was wound in a 7 67... closepack configuration to minimize the cross sectional area of the winding, which is reduced by a factor of 0.866 from a square winding pattern. The resulting winding crosssection is about 2.6 mm on a side, only about 6% of the coil radius, so the winding approximates a filament. The mean radius of the coils is 4.6 cm, 4.8 cm, and 5.1 cm. The Maxwell pair is wound on the outside because it is inherently more efficient and will hold down the other coils. The two halves of each coil are wound separately, and one channel of a stereo amplifier is wired to each. It is driven in current mode from the Techron 7540 amplifiers. The currentcontrol circuit helps to buck the inductance of the coil. The coil resistance for the radial windings is predicted to be about 6.7 Q for the inner set. The measured resistances and time constants including the leads and filters are given in Table 2. The time constant of the shorted power cable with filters and fuses was too short to measure with the amplifier, so it can be assumed to be negligible. The last column is the inductance estimated from the Bowtell and Mansfield formulation for coils on the surface of cylinders.72 To adapt for the thick winding, the height is added to the width of the winding. Calculations for loops of square cross section compare closely to heuristic formulas.73 Table for 9 Coil X1 X2 Y1 Y2 Zl Z2 2. Comparison of measured and predicted inductance cm gradient coil set. Measured Measured Time Experimental Theoretica Resistance Constant Inductance Inductanc (Q) (9s) (gH) (UH) 7.02 7.08 7.57 7.61 3.50 3.53 175 175 195 190 150 150 1229 1239 1476 1446 525 530 l e 1727 1727 1866 1866 330 330 The difference between the theoretical and experimental inductance is not due to the inductance of the power cable, which was measured by the same technique to be about 18 pH. It is primarily due to poor control of the crosssectional 72R. Bowtell and P. Mansfield, Meas. Sci. Technol. 1, 431, 1990. 73F. E. Terman, Radio Engineers' Handbook, McGrawHill, New York, 1943. dimensions of the winding, which expands when it is removed from the winding former. The maximum operating field as the system has been installed is 12.83 G/cm for X, 12.28 G/cm for Y and 8.80 G/cm for Z. By limiting the gradient field to a value below that corresponding to the steady state current, the switching time is better controlled. A drawing of the coil that illustrates how the Faraday shield and RF coil fit together is shown in Figure 22. Golay windings RF coil Faraday Centering Maxwell Clamping shield disk pair cam Figure 22. Drawing of 9 cm gradient coil set with Faraday shield and RF coil. Figure 23 is a photograph of the assembly. The former is an acrylic tube of 3.5" (89 mm) nominal outer diameter and 1/8" (3.2 mm) wall thickness. The radial gradient coils are wound on rectangular bobbins made from three flat pieces of acrylic. The wires are held together in a hexagonal matrix by General Electric varnish #7031 diluted in acetone. After winding, the bobbin is disassembled and the coil is glued onto the cylindrical former with epoxy. A variety of Figure 23. Photograph of 9 cm gradient coil set. The power cable and water supply cables are visible at left. The axial and radial gradient coils are visible through the cooling tubing. RF coils were developed as inserts for the probe, including 19F, 1H birdcage, 31P/1H doubletuned saddle coil, and my own 1H saddle coil. A pair of cams connected by a rod and mounted on the edge of the mounting flanges served to lock the probe into the magnet. A Faraday shield was used in addition to the filter/fuse box to isolate the RF coils from the gradient coils. The shield consisted of strips of Reynolds heavyduty aluminum foil approximately 2" wide, overlapped by about 1/2", and insulated from the other strips by masking tape that also secured the strips to manila card stock. At one end, all strips contacted a header strip. Provision was made to ground the shield, but in practice it was not used. The interdigitated geometry reduced eddy currents from the gradient coil, but allowed the shield to serve as a barrier to the RF field. The Maxwell pair is on the outside, which helps to hold the radial gradient coils in place. Epoxy was initially used to hold the windings to the former and pot the windings, but the epoxy did not withstand the temperatures developed by the coil and depolymerized. Polyester was selected as a casting resin that would outperform the acrylic former under warm conditions, and the coils were potted in polyester. In order to cool the unit, approximately 25 m of 1/8" O. D. by 1/64" wall polypropylene tubing was wound around the coils. It was connected to a Neslab circulating system that is capable of producing a pressure head of 40 psi. Supply tubing consisted of about 30 m of 1/4" 0. D. by 1/32" wall polypropylene. Assuming laminar flow, the water flux through a tube74 of radius r (cm) is flux = r 4Ap 8 l where Ap is the pressure in dyne/cm2, 1 is the length of the tube in cm, and g is the viscosity in poise. The resulting flux for a 40 psi drop is 8.6 cm3/s. One KW of power transferred to the water in the tube will raise its temperature by about 280 C. Measurement of the motion of bubbles in the tubing reveals a flow of about 3.5 cm3/s. 74D. Lide,(Ed.), CRC Handbook of Chemistry and Physics, 51st Edition, CRC Press, Boca Raton, 1970, p. F34. The difference is almost certainly due to quickrelease connectors that allow the probe to be removed or inserted at operating pressure. 15 cm Coil for Small Animals The 15 cm coil was designed to accommodate larger rats and other mediumsized laboratory animals and still produce a higher field and faster switching time than the Oxford gradient set. Like the 9 cm and the NMR microscope coil set, it is based on filamentary winding design. Since it is also driven by the Techron 7540 amplifier set, which is underpowered for a coil of this diameter, switching performance was at a premium. So, in contrast to the 9 cm coil, the 15 cm coil was designed to have optimal switching performance for the chosen switching time. In order to provide more flexibility in choosing either high field intensity or fast switching time, the windings were split and could be driven either in series or parallel. Since the coil would tend to become somewhat unwieldy as an insertable unit, its length was the shortest that would give essentially undiminished field intensity. Plots of the relative error in Figures 26 and 27 illustrate the linear region of the radial coil design. In order to avoid aliasing signals from long animals, the axial coil was based on an extended linearity design by Suits and Wilken.75 The 75B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989. coil and amplifiers were capable of developing 9 G/cm in a 100 gs switching time on X, Y and Z channels. The arcs in the 15 cm coil were arranged to have the minimum length without losing a significant amount of efficiency in the static limit. The standard solution of 68.70/21.30 for the arc positions arcl/arc2 leads to no third order component from either arc, but a family of solutions for which the third order components cancel is available. These solutions are graphed in Figure 24. arc2 (deg) 60 50 40 30 arcl (deg) 30 40 50 60 Figure 24. The solutions to the arc position of the double saddle radial gradient coil. Each solution is graphed twice, since exchanging arcl and arc2 results in the same coil design. To improve the relative size of the linear region to the coil, one would like to make the coil shorter than the 68.70/21.30 solution. The current efficiency decreases with the length, since the return arcs tend to cancel the desired field, and moving them closer increases the effect. However, in order to include the effect of reducing resistive loss in the coil, one must divide the current efficiency by the square root of the length. The resulting measure is an indication of the relative field that can be produced with an amplifier of a given power. Figure 25 illustrates the relative power efficiency as a function of the position of the return arc. rel. power eff. 0.3 0.25 0.2 0.15 0.1 0.05 angle (deg) 20 25 30 35 40 45 50 Figure 25. The relative power efficiency of the double saddle radial gradient coil as a function of the angle between the zaxis and the current return path. The peak efficiency is achieved at 260, but efficiency is a weak function of angle and much shorter coils can be used with little loss of performance. Based on this curve, the arcs of the 15 cm radial coil were placed at 30.20 and 66.10 from the axis of the coil, compared to 21.30 and 68.70 for the Golay coil. The overall length of the coil is reduced by 33%. The current efficiency is 0.819/a2 G/cmA compared to 0.808/a2 G/cmA for the 68.70/21.30 Golay coil. It would be even better to use a variant of the fieldversus switchingtime approach to determine the change in 0.4 1% 3% 0.2 1% Z 0 +1%+3%+5% 0.2 0.4 0.6 0.4 0.2 0 0.2 0.4 0.6 a) Y 0.4 +5% +3% +1% 0.2 Y 0 0.2 0.4 0.4 0.2 0 0.2 0.4 b) x Figure 26. Relative error plots of 30.20/66.10 radial gradient coil. Radius of coil corresponds to 1 on scale. a) YZ plane; b) XY plane. 1 0.61 +5% +3% 0.4 +1% 1% 5% 3% 0.2 Z 0 +1%+3%+5% 0.2 0.4 0.6 1 0.5 0 0.5 1 X Figure 26continued. Relative error plots of 30.20/66.10 radial gradient coil. Radius of coil corresponds to 1 on scale. c) XZ plane. performance for coils of different lengths, taking the changing inductance into account. The size and shape of the linear region of the 66.10 /30.20 radial gradient coil design is described by the plots of Figures 26 and 27. It is not greatly different from the longer 68.70/21.30 design. Note from Figure 26 that the regions of equal relative error are not simply connected. A threedimensional plot of a region as large as that in the two dimensional plots would give a false impression of the size of the linear region, since the apparently solid volumes would contain large holes. To avoid these "bubbles" of linearity and give a true picture of the useful volume, r the region displayed in Figure 27 is truncated. As a result of the truncation, it is possible to see through the linear region. All plots were produced by direct evaluation of the BiotSavart law. 2 z 0 1 0.5 x 0 0.5 1 0.5 0 0. Y 1 Figure 27. Perspective rendering of the 5% relative error region of 30.20/66.10 radial gradient coil. The axial coil was built to an extendedlinearity design proposed by Suits and Wilken.76 It consists of loops at both 40.00 and 66.30 from the z axis. The outer loops carry 7.5 times more current than the inner loops. Using the additional degrees of freedom of the secondloop position and the ratio of current in the loops, the fifth 76B. H. Suits and D. E. Wilken, J. Phys. E: Sci. Instrum. 23, 565, 1989. and seventh order terms are canceled, resulting in about eight times the useful volume of a Maxwell pair. For a coil with more turns in some loops than others, the current efficiency does not have an unambiguous definition. With respect to the current in the outer loops, the current efficiency of the coil is 0.635/a2 G/cmA. The 15 cm coil was matched to the Techron 7540 amplifiers in our laboratory using the timedomain model of gradient performance described above. The height and width of the windings was set to be 1 by 1 cm for the radial and 1 by 2 cm for the outer axial, to ensure that the assumption of filamentary wires in the calculation of angle position would be valid. The width of the outer axial winding was increased from 1 to 2 cm, since it is farther from the center than the others, and it was necessary to increase it to match the radial performance. Inductance of the radial and axial coils was calculated using a FourierBessel approach.77 The available combinations of switching time and maximum gradient are shown in Figure 28. In order to improve the switching time to smaller fields, the 15 cm coil is constructed from split windings. All the gradient coils described have the two sides driven by separate amplifiers so that the magnetic center of the coil can be moved. The 15 cm coil has each side split further into two identical but electrically separate 77R. Bowtell and P. Mansfield, Meas. Sci. Technol. 1, 431, 1990. windings. Placing the windings in parallel trades field intensity for switching time; placing them in series reduces switching time at the expense of lower field intensity. Gma (G/cm) 14 X 12 10 8 6 4 2 to (ms) 0.5 1 1.5 2 Figure 28. Maximum field Gmax (G/cm) vs. switching time to (ms) for the 15 cm field gradient coil set as driven by the Techron 7540. Points along the curves represent designs with different numbers of turns, increasing from left to right. The top curve represents the inner radial coil. The lower curves represent the outer radial coil and the axial coil. A "slower" coil has a larger maximum field. All three coils are optimized to approximately 10 G/cm for the series mode. The exact number of turns for the desired field is not used, due to the limited number of standard wire sizes and the use of rectangular winding cross sections. For the number of turns actually used, to and Gmax based on the time domain model are tabulated in Table 3. A photograph of the 15 cm coil assembly is shown in Figure 29. The coil is wound on an acrylic former having a nominal O. D. of 6" (152 mm) and a wall thickness of 1/4" (6.4 mm). The overall length was 38 mm. The radial coils 
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